[{"data":1,"prerenderedAt":1134},["ShallowReactive",2],{"layer:shape-and-space:understand":3},{"layer":4,"contentHash":1115,"dependencyHashes":1116,"approval":1127,"releaseId":1133},{"schemaVersion":5,"conceptId":6,"locale":7,"depth":8,"revision":5,"title":9,"subtitle":10,"summary":11,"objectives":12,"estimatedMinutes":18,"plate":19,"blocks":40,"sourceIds":1110,"reviewStatus":1111,"authoring":1112},1,"shape-and-space","en","understand","Naming shapes precisely","Definitions, properties and the mix-ups they clear up","Give every shape an exact definition: polygons and diagonals, triangles by sides and angles, the quadrilateral family tree, the parts of a circle, perimeter, prisms and pyramids, nets, views and line symmetry, with worked examples and common mix-ups.",[13,14,15,16,17],"Use exact definitions to decide whether a figure is a polygon, and whether it is regular.","Count the diagonals of a polygon and classify triangles by both sides and angles.","Place any quadrilateral in the family tree and use side, angle and diagonal properties to name it.","Name the parts of a circle, calculate perimeters and circumferences, and count F, E and V for prisms and pyramids.","Match solids to their nets and views, and find the lines of symmetry of polygons and letters.",40,{"title":20,"rows":21},"Lesson plate",[22,25,28,31,34,37],{"label":23,"value":24},"Depth","Understand",{"label":26,"value":27},"Reading time","≈ 40 minutes",{"label":29,"value":30},"Prior knowledge","Discover layer; angle words",{"label":32,"value":33},"Chapters","10",{"label":35,"value":36},"Labs","Triangle and quadrilateral sorts, F\u002FE\u002FV count and match",{"label":38,"value":39},"Key numbers","180°, 3.14, n + 2, 3n, 2n",[41,45,51,57,60,88,93,96,101,104,118,150,155,170,175,178,201,206,216,287,292,298,303,306,345,371,375,455,459,484,489,492,531,535,551,563,575,580,585,588,603,612,616,621,624,652,661,682,714,719,724,727,759,763,777,782,785,794,798,803,806,830,849,853,864,869,950,1070,1074,1091,1095,1100],{"id":42,"type":43,"markdown":44},"intro-precise","prose","In Discover you met shapes the friendly way: by looking, touching and folding. Now we sharpen the words. In mathematics a name is a **promise**: if someone says a shape is a rhombus, you know exactly which facts are guaranteed and which are not.\n\nThat precision is what lets a carpenter in Jaipur and an engineer in Chennai talk about the same shape without a picture, and it is what lets us answer questions like *is a square a rectangle?* without arguing. This layer gives each shape a proper **definition** (the smallest set of facts that decides whether something belongs to the family), lists its **properties** (everything else that follows), and clears up the mix-ups that trip most learners.",{"id":46,"type":47,"variant":48,"title":49,"markdown":50},"intro-def-vs-prop","callout","definition","Definition versus property","A **definition** is the test for membership: *a rectangle is a quadrilateral with four right angles.* A **property** is something that is then always true: *the diagonals of a rectangle are equal.* You check the definition to name a shape; you use properties to solve problems about it. Many mix-ups come from treating a property as if it were the definition.",{"id":52,"type":53,"title":54,"eyebrow":55,"navLabel":56},"ch01","chapter","From curves to polygons","Chapter 01","1 Curves and polygons",{"id":58,"type":43,"markdown":59},"poly-precise","A **line segment** is the straight path between two points. A **polygon** is a **simple closed** figure made **only** of line segments, where:\n\n1. each segment meets exactly two others, one at each end;\n2. the segments meet only at their endpoints (no crossing);\n3. the figure closes up.\n\nThe segments are **sides**, the meeting points are **vertices**, and at each vertex the two sides make an **interior angle**. Two sides that share a vertex are **adjacent sides**; two vertices at the ends of one side are **adjacent vertices**. The fewest sides possible is 3, because two segments cannot enclose anything.",{"id":61,"type":62,"caption":63,"columns":64,"rows":70},"table-is-polygon","table","Polygon or not? Test each rule",[65,66,67,68,69],"Figure","Closed?","Straight sides only?","Simple (no crossing)?","Polygon?",[71,75,78,80,82,85],[72,73,73,73,74],"Triangle","Yes","**Yes**",[76,73,77,73,77],"Circle","No",[79,77,73,73,77],"Letter Z drawn with 3 strokes",[81,73,73,77,77],"Five-pointed star drawn in one go (lines crossing)",[83,73,73,73,84],"Star outline (10 sides, no crossing)","**Yes**, a decagon",[86,73,87,73,77],"Semicircle","No, one side is an arc",{"id":89,"type":47,"variant":90,"title":91,"markdown":92},"aha-star","aha","A star can be a decagon","Trace just the **outline** of a five-pointed star, the way you would cut it from paper. It has 5 outer points and 5 inner corners, so it has **10 sides** and **10 vertices**: it is a (non-regular) **decagon**. Some of its angles point inwards, bigger than a straight angle. A polygon like this, with a dent in it, is called **concave**. A polygon with no dents, where every diagonal stays inside, is **convex**.",{"id":94,"type":43,"markdown":95},"regular-precise","A **regular polygon** has **all sides equal and all angles equal**. Both conditions matter:\n\n- A **rhombus** that is not a square has equal sides but unequal angles, so it is not regular.\n- A **rectangle** that is not a square has equal angles but unequal sides, so it is not regular.\n- Only the **square** has both, so the square is the one regular quadrilateral.\n\nFor triangles something special happens: if all three sides are equal, the angles are automatically equal too (each 60°). So an equilateral triangle is always regular.",{"id":97,"type":53,"title":98,"eyebrow":99,"navLabel":100},"ch02","Sides, vertices, angles and diagonals","Chapter 02","2 Diagonals",{"id":102,"type":43,"markdown":103},"diag-meet","A **diagonal** is a line segment joining two vertices that are **not** next to each other. Sides join neighbouring vertices; diagonals join the others.\n\nA triangle has **no** diagonals: every vertex is next to both of the others. A quadrilateral has **2**. A pentagon has **5**, which together draw the famous five-pointed star inside it. As the number of sides grows, the diagonals multiply quickly.",{"id":105,"type":106,"title":107,"problem":108,"steps":109,"help":116},"we-diag-hex","worked_example","Counting the diagonals of a hexagon","How many diagonals does a hexagon have? Count them without missing any or counting any twice.",[110,111,112,113,114,115],"Label the vertices A, B, C, D, E, F going round.","From A, you cannot draw a diagonal to A itself or to its neighbours B and F. You can draw to C, D and E: **3 diagonals**.","Every vertex is the same: 3 diagonals start at each of the 6 vertices, giving 6 × 3 = 18.","But the diagonal AC was counted from A **and** again from C. Every diagonal has been counted **twice**.","So the hexagon has 18 ÷ 2 = **9 diagonals**.","Check by listing: AC, AD, AE, BD, BE, BF, CE, CF, DF. That is 9. ✓",{"simplerExplanation":117},"Each corner connects to 3 far corners. 6 corners × 3 = 18, but each line has two ends, so halve it: 9.",{"id":119,"type":62,"caption":120,"columns":121,"rows":126},"table-diag-small","Diagonals of small polygons (by listing)",[122,123,124,125],"Polygon","Sides","Diagonals from one vertex","Total diagonals",[127,130,135,138,142,146],[72,128,129,129],"3","0",[131,132,133,134],"Quadrilateral","4","1","2",[136,137,134,137],"Pentagon","5",[139,140,128,141],"Hexagon","6","9",[143,144,132,145],"Heptagon","7","14",[147,148,137,149],"Octagon","8","20",{"id":151,"type":47,"variant":152,"title":153,"markdown":154},"obs-diag-rule","observation","A rule is hiding here","From each vertex you can reach every other vertex except itself and its 2 neighbours: that is **n − 3** diagonals. Multiply by the n vertices and halve for double counting. In the Deepen layer you will turn this into the formula n(n − 3) ÷ 2 and prove it always works.",{"id":156,"type":157,"itemId":158,"prompt":159,"check":160,"hints":164,"feedback":167},"pr-oct-diag","practice","shape-and-space.understand-octagon-diagonals","How many diagonals does an **octagon** (8 sides) have?",{"kind":161,"answer":162,"tolerance":163},"number",20,0,[165,166],"How many diagonals leave each vertex? (All vertices except itself and its 2 neighbours.)","Multiply by 8, then halve.",{"correct":168,"incorrect":169},"Yes: 5 from each vertex, 8 × 5 = 40, and 40 ÷ 2 = 20.","From each vertex, 8 − 3 = 5 diagonals. 8 × 5 = 40 counts every diagonal twice, so the answer is 40 ÷ 2 = 20.",{"id":171,"type":53,"title":172,"eyebrow":173,"navLabel":174},"ch03","Triangles, sorted two ways","Chapter 03","3 Triangles",{"id":176,"type":43,"markdown":177},"tri-two-ways","Every triangle can be described in **two** separate ways at once: by its **sides** and by its **angles**.\n\n**By sides:** equilateral (all three equal), isosceles (two equal), scalene (all different).\n\n**By angles:** acute-angled (all three angles less than 90°), right-angled (one angle exactly 90°), obtuse-angled (one angle more than 90°).\n\nSo a full description sounds like *a right-angled isosceles triangle* (the shape of a folded square scarf) or *an obtuse-angled scalene triangle*.",{"id":179,"type":62,"caption":180,"columns":181,"rows":186},"table-tri-combos","Which side-and-angle combinations are possible?",[182,183,184,185],"By sides ↓ \u002F by angles →","Acute-angled","Right-angled","Obtuse-angled",[187,191,196],[188,189,190,190],"Equilateral","Yes (always: every angle is 60°)","Impossible",[192,193,194,195],"Isosceles","Yes (e.g. 70°, 70°, 40°)","Yes (45°, 45°, 90°)","Yes (e.g. 30°, 30°, 120°)",[197,198,199,200],"Scalene","Yes (e.g. 50°, 60°, 70°)","Yes (e.g. 30°, 60°, 90°)","Yes (e.g. 20°, 40°, 120°)",{"id":202,"type":47,"variant":203,"title":204,"markdown":205},"careful-one-angle","careful","A triangle can have at most one right or obtuse angle","The three angles add up to exactly **180°**. Two right angles would already use 90° + 90° = 180°, leaving 0° for the third corner, which is impossible. The same goes for two obtuse angles (more than 180° already). So every triangle has **at least two acute angles**, and you classify it by its **largest** angle.",{"id":207,"type":106,"title":208,"problem":209,"steps":210},"we-missing-angle","Finding the third angle","A triangle has angles of 48° and 67°. Find the third angle and classify the triangle by its angles.",[211,212,213,214,215],"The three angles of a triangle add up to 180°.","Known angles: 48° + 67° = 115°.","Third angle: 180° − 115° = **65°**.","All three angles (48°, 67°, 65°) are less than 90°, so it is an **acute-angled** triangle.","The angles are all different, so the sides are all different too: it is also **scalene**.",{"id":217,"type":218,"component":219,"componentVersion":5,"config":220,"objective":281,"textAlternative":282,"help":283},"lab-sort-triangles","interactive","sort-game",{"prompt":221,"bins":222,"items":232,"seconds":163},"Each card gives the three angles of a triangle. Sort by the kind of triangle.",[223,225,227,229],{"id":224,"label":183},"acute",{"id":226,"label":184},"right",{"id":228,"label":185},"obtuse",{"id":230,"label":231},"impossible","Not a triangle",[233,237,241,245,249,253,257,261,265,269,273,277],{"id":234,"label":235,"bin":224,"why":236},"t1","60°, 60°, 60°","All three are less than 90°. This is the equilateral triangle.",{"id":238,"label":239,"bin":226,"why":240},"t2","90°, 45°, 45°","One angle is exactly 90°. It is also isosceles: half of a square cut along a diagonal.",{"id":242,"label":243,"bin":228,"why":244},"t3","120°, 30°, 30°","120° is more than 90°, so it is obtuse-angled (and isosceles).",{"id":246,"label":247,"bin":226,"why":248},"t4","90°, 60°, 30°","One right angle. The other two add to 90°. This is the shape of a set square.",{"id":250,"label":251,"bin":224,"why":252},"t5","80°, 55°, 45°","Adds to 180° and every angle is under 90°.",{"id":254,"label":255,"bin":228,"why":256},"t6","100°, 45°, 35°","Adds to 180°, and 100° is obtuse.",{"id":258,"label":259,"bin":230,"why":260},"t7","90°, 90°, 10°","These add to 190°, not 180°. Two right angles leave no room for a third corner.",{"id":262,"label":263,"bin":230,"why":264},"t8","70°, 60°, 40°","70 + 60 + 40 = 170°, not 180°, so no triangle has these angles.",{"id":266,"label":267,"bin":224,"why":268},"t9","89°, 46°, 45°","89° is just under a right angle, so every angle is acute.",{"id":270,"label":271,"bin":228,"why":272},"t10","91°, 45°, 44°","91° is just over 90°: obtuse, even though it looks almost right-angled.",{"id":274,"label":275,"bin":230,"why":276},"t11","30°, 30°, 30°","Only 90° in total. Every triangle needs exactly 180°.",{"id":278,"label":279,"bin":226,"why":280},"t12","25°, 65°, 90°","Adds to 180° with one right angle.","Classify triangles by their angles, and spot angle sets that cannot make a triangle at all.","Twelve cards each give three angles. The four bins are acute-angled, right-angled, obtuse-angled and not a triangle.\n\nFirst check the total is 180°. Three cards fail: 90°, 90°, 10° (190°), 70°, 60°, 40° (170°) and 30°, 30°, 30° (90°). They go in **not a triangle**.\n\nThen look at the largest angle. Right-angled: 90°, 45°, 45°; 90°, 60°, 30°; 25°, 65°, 90°. Obtuse-angled: 120°, 30°, 30°; 100°, 45°, 35°; 91°, 45°, 44°. Acute-angled: 60°, 60°, 60°; 80°, 55°, 45°; 89°, 46°, 45°.\n\nThe close calls, 89° and 91°, show that the name depends on the exact angle, not on how the triangle looks.",{"hints":284},[285,286],"Always add the three angles first. Not 180°? Not a triangle.","Then only the biggest angle matters.",{"id":288,"type":47,"variant":289,"title":290,"markdown":291},"misc-isosceles-look","misconception","“An isosceles triangle must point upwards”","Textbook pictures usually show isosceles triangles standing on their odd side with the point at the top. Turn one on its side and it is still isosceles: the test is **two equal sides**, not the direction it faces. Likewise, a right-angled triangle does not need its right angle at the bottom left.",{"id":293,"type":294,"conceptId":295,"relation":296,"explanation":297},"conn-angles-und","connection","angles","helps_understand","Classifying triangles needs the angle words acute, right and obtuse, and the angle-sum fact (180°) comes from angles on a straight line.",{"id":299,"type":53,"title":300,"eyebrow":301,"navLabel":302},"ch04","The quadrilateral family tree","Chapter 04","4 Quadrilaterals",{"id":304,"type":43,"markdown":305},"quad-defs","Here are the definitions used in Indian school textbooks. Each is the **smallest** test; the table after it lists the properties that follow.\n\n- **Trapezium:** a quadrilateral with **a pair of parallel sides**.\n- **Parallelogram:** a quadrilateral with **both pairs** of opposite sides parallel.\n- **Rhombus:** a parallelogram with **all four sides equal** (equivalently, any quadrilateral with four equal sides).\n- **Rectangle:** a parallelogram with **a right angle** (then all four angles are right angles).\n- **Square:** a rectangle with **all sides equal** (equivalently, a rhombus with a right angle).\n- **Kite:** a quadrilateral with **two pairs of equal sides next to each other** (adjacent), such as AB = AD and CB = CD.",{"id":307,"type":62,"caption":308,"columns":309,"rows":314},"table-quad-props","Properties of the quadrilateral family",[310,123,311,312,313],"Shape","Angles","Diagonals","Lines of symmetry",[315,320,324,329,334,340],[316,317,318,319,132],"Square","All 4 equal; opposite sides parallel","All 90°","Equal; bisect each other at 90°",[321,322,318,323,134],"Rectangle","Opposite sides equal and parallel","Equal; bisect each other",[325,317,326,327,328],"Rhombus","Opposite angles equal","Bisect each other at 90°; not equal (unless a square)","2 (the diagonals)",[330,322,331,332,333],"Parallelogram","Opposite angles equal; neighbours add to 180°","Bisect each other","0 (unless it is a rectangle or rhombus)",[335,336,337,338,339],"Trapezium","One pair of opposite sides parallel","Angles along each slanting side add to 180°","Nothing special in general","0, or 1 if isosceles",[341,342,343,344,133],"Kite","Two pairs of adjacent sides equal","One pair of opposite angles equal","Meet at 90°; one bisects the other",{"id":346,"type":347,"title":348,"items":349},"steps-family-tree","steps","The family tree, from general to special",[350,353,356,359,362,365,368],{"title":131,"tag":351,"text":352},"any 4 sides","Every shape below is one of these.",{"title":335,"tag":354,"text":355},"+ one pair parallel","Add one pair of parallel sides.",{"title":330,"tag":357,"text":358},"+ both pairs parallel","Both pairs parallel. Opposite sides and angles become equal.",{"title":321,"tag":360,"text":361},"+ a right angle","A parallelogram with a right angle. All angles become 90°.",{"title":325,"tag":363,"text":364},"+ all sides equal","A parallelogram with 4 equal sides. Rectangle and rhombus are sister branches.",{"title":316,"tag":366,"text":367},"rectangle AND rhombus","Both a rectangle and a rhombus at once: right angles and equal sides.",{"title":341,"tag":369,"text":370},"a separate branch","Adjacent pairs equal. A rhombus is a kite too (all four sides equal means both adjacent pairs match).",{"id":372,"type":47,"variant":90,"title":373,"markdown":374},"aha-square-everything","The square belongs to five families","Check each definition in turn. A square has a pair of parallel sides (trapezium ✓), both pairs parallel (parallelogram ✓), a right angle (rectangle ✓), four equal sides (rhombus ✓) and two pairs of equal adjacent sides (kite ✓). The square is the most special quadrilateral of all, and it inherits **every** property of every family above it.",{"id":376,"type":218,"component":219,"componentVersion":5,"config":377,"objective":448,"textAlternative":449,"help":450},"lab-sort-quads",{"prompt":378,"bins":379,"items":391,"seconds":163},"Give each description its most special name.",[380,382,384,386,388],{"id":381,"label":316},"square",{"id":383,"label":321},"rectangle",{"id":385,"label":325},"rhombus",{"id":387,"label":330},"parallelogram",{"id":389,"label":390},"other","Trapezium or kite",[392,396,400,404,408,412,416,420,424,428,432,436,440,444],{"id":393,"label":394,"bin":381,"why":395},"q1","Four equal sides and one right angle","Equal sides make it a rhombus; a right angle as well makes it a square.",{"id":397,"label":398,"bin":383,"why":399},"q2","Opposite sides parallel, all angles 90°, sides 8 cm and 5 cm","Right angles but unequal sides: a rectangle that is not a square.",{"id":401,"label":402,"bin":385,"why":403},"q3","Four sides of 6 cm, angles 60° and 120°","Equal sides but no right angle: a rhombus, not a square.",{"id":405,"label":406,"bin":387,"why":407},"q4","Opposite sides parallel, sides 7 cm and 4 cm, angles 70° and 110°","Both pairs parallel, but neither right angles nor equal sides.",{"id":409,"label":410,"bin":389,"why":411},"q5","Only one pair of opposite sides parallel","That is a trapezium (the other pair is not parallel).",{"id":413,"label":414,"bin":389,"why":415},"q6","AB = AD = 5 cm and CB = CD = 9 cm","Two pairs of equal adjacent sides: a kite.",{"id":417,"label":418,"bin":381,"why":419},"q7","Diagonals equal and bisecting each other at right angles","Equal diagonals give a rectangle; perpendicular bisecting diagonals give a rhombus. Both at once: a square.",{"id":421,"label":422,"bin":383,"why":423},"q8","Diagonals equal and bisecting each other, but not at right angles","Bisecting diagonals make a parallelogram; equal diagonals make it a rectangle.",{"id":425,"label":426,"bin":385,"why":427},"q9","Diagonals bisect each other at 90° but are of different lengths","Perpendicular bisecting diagonals mean a rhombus; unequal diagonals mean it is not a square.",{"id":429,"label":430,"bin":387,"why":431},"q10","Diagonals bisect each other, unequal, not at right angles","Bisecting diagonals is the parallelogram test; nothing extra is given.",{"id":433,"label":434,"bin":381,"why":435},"q11","A carrom board","Four equal sides and four right angles.",{"id":437,"label":438,"bin":383,"why":439},"q12","A cricket pitch (20.12 m by 3.05 m)","Right-angled corners, long and narrow.",{"id":441,"label":442,"bin":389,"why":443},"q13","The side view of a bucket","The rim and base are parallel; the slanted sides are not: a trapezium.",{"id":445,"label":446,"bin":389,"why":447},"q14","A patang whose top sticks are shorter than its bottom sticks","Adjacent pairs are equal but the pairs differ: a kite.","Use the definitions to give every quadrilateral its most special name, including from facts about its diagonals.","Fourteen cards describe a quadrilateral, either by its sides and angles, by its diagonals or as an everyday object. The five bins are square, rectangle, rhombus, parallelogram, and trapezium or kite.\n\n**Square:** four equal sides and a right angle; diagonals equal and bisecting at right angles; a carrom board.\n**Rectangle:** right angles with sides 8 cm and 5 cm; diagonals equal and bisecting but not at right angles; a cricket pitch.\n**Rhombus:** four 6 cm sides with angles 60° and 120°; diagonals bisecting at 90° but unequal.\n**Parallelogram:** sides 7 cm and 4 cm with angles 70° and 110°; diagonals bisecting but unequal and not perpendicular.\n**Trapezium or kite:** only one pair of parallel sides; AB = AD and CB = CD; the side view of a bucket; a patang with unequal pairs of sticks.\n\nThe rule: always give the **most special** name that the facts guarantee.",{"simplerExplanation":451,"hints":452},"Start general and add conditions. Parallel both ways? Parallelogram. Add a right angle: rectangle. Add equal sides: rhombus. Add both: square.",[453,454],"Diagonals that bisect each other always mean a parallelogram (or something more special).","Equal diagonals point to a rectangle; perpendicular ones point to a rhombus.",{"id":456,"type":47,"variant":289,"title":457,"markdown":458},"misc-rect-not-square","“A rectangle has two long and two short sides”","That is only how rectangles are usually **drawn**. The definition asks for four right angles and nothing about side lengths. If all four sides happen to be equal, the shape is still a rectangle; it is just the special rectangle we also call a square. The same logic makes every square a rhombus, and every rhombus a parallelogram.",{"id":460,"type":157,"itemId":461,"prompt":462,"check":463,"hints":479,"feedback":481},"pr-quad-true","shape-and-space.understand-quad-true","Which statement is **false**?",{"kind":464,"options":465,"correct":478},"choice",[466,469,472,475],{"id":467,"label":468},"a","Every rhombus is a parallelogram",{"id":470,"label":471},"b","Every square is a rhombus",{"id":473,"label":474},"c","Every rectangle is a square",{"id":476,"label":477},"d","Every rectangle is a parallelogram",[473],[480],"Look for a rectangle that fails the square test.",{"correct":482,"incorrect":483},"Correct. A 6 cm by 4 cm rectangle has right angles but unequal sides, so it is not a square.","The false one is “every rectangle is a square”. A rectangle needs only right angles; its sides can be different, like a door.",{"id":485,"type":53,"title":486,"eyebrow":487,"navLabel":488},"ch05","The parts of a circle","Chapter 05","5 Circles",{"id":490,"type":43,"markdown":491},"circle-precise","A **circle** is the set of all points in a plane that are the same distance from a fixed point, the **centre**. That distance is the **radius** (r).\n\nThe words for the parts of a circle are worth learning exactly, because they appear in engineering, astronomy and sport:",{"id":493,"type":62,"caption":494,"columns":495,"rows":499},"table-circle-parts","Parts of a circle",[496,497,498],"Part","Meaning","Example",[500,504,508,512,516,520,524,528],[501,502,503],"Radius","A segment from the centre to a point on the circle, or its length","A spoke of a bicycle wheel",[505,506,507],"Diameter","A chord through the centre; its length is 2 × radius","The width of a round roti measured straight across the middle",[509,510,511],"Chord","A segment joining any two points on the circle","A straight cut across a pizza that misses the centre",[513,514,515],"Arc","A piece of the circle between two points","The curved crust of a pizza slice",[517,518,519],"Circumference","The whole distance round the circle","The length of a bangle if you cut it and straighten it",[521,522,523],"Sector","The region between two radii and the arc: a “slice”","A slice of a round cake cut from the centre",[525,526,527],"Segment","The region between a chord and its arc","The piece left when you cut straight across a roti",[86,529,530],"Half a circle, cut off by a diameter","The “D” of a hockey or football goal area",{"id":532,"type":47,"variant":90,"title":533,"markdown":534},"aha-diameter-longest","The diameter is the longest chord","Every chord joins two points of the circle. The chord that passes through the centre, the diameter, is the longest possible one. Slide a chord away from the centre and it gets shorter, until at the very edge it shrinks to a single point.",{"id":536,"type":537,"items":538},"formulas-circle","formulas",[539,542,545,548],{"expression":540,"caption":541},"d = 2 × r","Diameter is twice the radius.",{"expression":543,"caption":544},"r = d ÷ 2","Radius is half the diameter.",{"expression":546,"caption":547},"C ≈ 3.14 × d","Circumference is a little more than 3 times the diameter. The exact number is called π (pi).",{"expression":549,"caption":550},"C = 2 × π × r","The same rule written with the radius, since d = 2r.",{"id":552,"type":106,"title":553,"problem":554,"steps":555,"help":561},"we-wheel","How far does a bicycle wheel roll in one turn?","A bicycle wheel has a diameter of 70 cm. How far does the cycle move forward when the wheel turns once? About how many turns does it make in 1 km?",[556,557,558,559,560],"In one turn, every point of the rim touches the road once, so the cycle moves forward by exactly **one circumference**.","C ≈ 3.14 × 70 = **219.8 cm**, about 2.2 m.","1 km = 1,000 m = 100,000 cm.","Turns ≈ 100,000 ÷ 219.9 ≈ **455 turns** (using a more exact value of π).","Check: 455 × 2.2 m ≈ 1,001 m. ✓",{"simplerExplanation":562},"One turn of the wheel lays its whole rim down on the road once. So the distance per turn is the rim length, a bit over 3 times 70 cm.",{"id":564,"type":157,"itemId":565,"prompt":566,"check":567,"hints":570,"feedback":572},"pr-radius","shape-and-space.understand-radius","A round clock face has a diameter of 30 cm. What is its radius, in cm?",{"kind":161,"answer":568,"tolerance":163,"unit":569},15,"cm",[571],"The radius is half of the diameter.",{"correct":573,"incorrect":574},"Yes: 30 ÷ 2 = 15 cm.","The diameter goes all the way across; the radius goes only from the centre to the edge, which is half: 30 ÷ 2 = 15 cm.",{"id":576,"type":294,"conceptId":577,"relation":578,"explanation":579},"conn-constructing-und","constructing-angles","related_to","A compass keeps a fixed radius, so it draws circles and marks equal lengths, the basis of every ruler-and-compass construction.",{"id":581,"type":53,"title":582,"eyebrow":583,"navLabel":584},"ch06","Perimeter: the distance around","Chapter 06","6 Perimeter",{"id":586,"type":43,"markdown":587},"perimeter","The **perimeter** of a closed figure is the total length of its boundary: the distance an ant walks to go once all the way round. For a polygon, just add the lengths of all the sides. For a circle, the perimeter has its own name: the circumference.\n\nPerimeter answers questions like: *how much fencing for this garden? how much lace for the edge of this dupatta? how far is one lap of the school ground?*",{"id":589,"type":537,"items":590},"formulas-perimeter",[591,594,597,600],{"expression":592,"caption":593},"P = 4 × side","Square (all four sides equal).",{"expression":595,"caption":596},"P = 2 × (length + breadth)","Rectangle: two lengths and two breadths.",{"expression":598,"caption":599},"P = n × side","Any regular polygon with n equal sides.",{"expression":601,"caption":602},"P = sum of all sides","Any polygon, regular or not.",{"id":604,"type":106,"title":605,"problem":606,"steps":607},"we-fence","Fencing three gardens","Find the perimeter of (a) a square plot of side 12 m, (b) a rectangular kitchen garden 15 m by 8 m, (c) a regular hexagonal flower bed of side 5 m.",[608,609,610,611],"(a) Square: 4 × 12 = **48 m** of fencing.","(b) Rectangle: 2 × (15 + 8) = 2 × 23 = **46 m**.","(c) Regular hexagon: 6 equal sides, so 6 × 5 = **30 m**.","Notice (a) and (b) have very different shapes but similar perimeters. Perimeter measures only the boundary, not how much ground is inside (that is **area**).",{"id":613,"type":47,"variant":289,"title":614,"markdown":615},"misc-perimeter-area","“Same perimeter means same size”","A 10 m by 2 m rectangle and a 6 m by 6 m square both have a perimeter of 24 m. But the rectangle covers 20 square metres and the square covers 36. Perimeter is the length of the fence; area is the ground inside it. Two plots with the same fence can hold very different numbers of plants.",{"id":617,"type":53,"title":618,"eyebrow":619,"navLabel":620},"ch07","Solids: faces, edges and vertices","Chapter 07","7 Solids",{"id":622,"type":43,"markdown":623},"solid-precise","A **polyhedron** (plural *polyhedra*) is a solid whose surface is made entirely of **flat polygons**. Its **faces** are those polygons, its **edges** are the segments where two faces meet, and its **vertices** are the points where three or more edges meet. Cubes, cuboids, prisms and pyramids are polyhedra. Cylinders, cones and spheres are **not**, because they have curved surfaces.\n\nTwo big families of polyhedra are named after the shape of their **base**:\n\n- A **prism** has two identical, parallel ends (the bases) joined by rectangles. A prism with triangle ends is a *triangular prism*; with hexagon ends, a *hexagonal prism*. A cuboid is a rectangular prism, and a cube is a square prism with square sides too.\n- A **pyramid** has one base, and triangles rising from every side of the base to a single point called the **apex**. A pyramid with a square base is a *square pyramid*; with a triangle base, a *triangular pyramid* (or **tetrahedron**).",{"id":625,"type":62,"caption":626,"columns":627,"rows":632},"table-prisms-pyramids","Faces, edges and vertices of prisms and pyramids",[628,629,630,631],"Solid","Faces","Edges","Vertices",[633,635,638,641,644,646,648,650],[634,137,141,140],"Triangle prism",[636,140,637,148],"Square prism (cuboid)","12",[639,144,640,33],"Pentagon prism","15",[642,148,643,637],"Hexagon prism","18",[645,132,140,132],"Triangle pyramid",[647,137,148,137],"Square pyramid",[649,140,33,140],"Pentagon pyramid",[651,144,637,144],"Hexagon pyramid",{"id":653,"type":106,"title":654,"problem":655,"steps":656},"we-hex-prism","Counting a hexagonal prism","A new pencil (before sharpening) is a hexagonal prism. Count its faces, edges and vertices.",[657,658,659,660],"Faces: the 2 hexagon ends plus one rectangle for each of the 6 sides: 2 + 6 = **8 faces**.","Edges: 6 round the top hexagon, 6 round the bottom hexagon, and 6 running along the length: 6 + 6 + 6 = **18 edges**.","Vertices: 6 at the top end and 6 at the bottom end: **12 vertices**.","Pattern: for a prism with an n-sided base, faces = n + 2, edges = 3n, vertices = 2n. With n = 6: 8, 18, 12. ✓",{"id":662,"type":218,"component":663,"componentVersion":5,"config":664,"objective":676,"textAlternative":677,"help":678},"lab-count-prisms","shape-explorer",{"solids":665,"polygons":672,"modes":673},[666,667,668,669,670,671],"triangular-prism","cube","pentagonal-prism","hexagonal-prism","triangular-pyramid","square-pyramid",[],[674,675],"count","explore","Type the faces, edges and vertices of prisms and pyramids, then explore them to check and find the pattern in each family.","Six solids: three prisms (triangular, pentagonal, hexagonal), a cube (a square prism) and two pyramids (triangular and square).\n\nIn **count**, each round shows a solid and you type F, E and V. In **explore**: Drag the solid, or use the Turn and Tilt sliders, to rotate it; hidden edges show as dashed lines. You can highlight its faces, edges or vertices, and for a polyhedron the Net button unfolds it flat. It also shows the counts and the Euler check.\n\nPrisms: triangular 5 faces, 9 edges, 6 vertices; cube 6, 12, 8; pentagonal 7, 15, 10; hexagonal 8, 18, 12. Each extra side on the base adds 1 face, 3 edges and 2 vertices. Pyramids: triangular 4, 6, 4; square 5, 8, 5. Faces and vertices are equal, and edges are twice the base sides.\n\nFor every one, F + V − E = **2**: Euler's formula, which you will meet again.",{"hints":679},[680,681],"Count the base, then the other end or apex, then what joins them.","Try faces + vertices − edges for each one.",{"id":683,"type":218,"component":684,"componentVersion":5,"config":685,"objective":709,"textAlternative":710,"help":711},"lab-match-fev","match-pairs",{"prompt":686,"mode":687,"pairs":688},"Match each solid with its faces (F), edges (E) and vertices (V).","connect",[689,692,695,697,700,703,706],{"a":690,"b":691},"Cube","F 6, E 12, V 8",{"a":693,"b":694},"Triangular prism","F 5, E 9, V 6",{"a":647,"b":696},"F 5, E 8, V 5",{"a":698,"b":699},"Triangular pyramid (tetrahedron)","F 4, E 6, V 4",{"a":701,"b":702},"Pentagonal prism","F 7, E 15, V 10",{"a":704,"b":705},"Hexagonal prism","F 8, E 18, V 12",{"a":707,"b":708},"Hexagonal pyramid","F 7, E 12, V 7","Connect each solid to its correct face, edge and vertex counts.","Seven solids on the left, seven sets of counts on the right. The correct pairs are: cube with 6 faces, 12 edges, 8 vertices; triangular prism with 5, 9, 6; square pyramid with 5, 8, 5; triangular pyramid with 4, 6, 4; pentagonal prism with 7, 15, 10; hexagonal prism with 8, 18, 12; hexagonal pyramid with 7, 12, 7.\n\nA quick way to tell them apart: pyramids have equal numbers of faces and vertices; prisms have twice as many vertices as base sides. Watch the two with 5 faces: the triangular prism (9 edges) and the square pyramid (8 edges).",{"hints":712},[713],"Pyramids: faces = vertices. Prisms: vertices = 2 × base sides.",{"id":715,"type":47,"variant":716,"title":717,"markdown":718},"nuance-curved-count","nuance","Counting curved solids needs care","For a cylinder, most Indian textbooks say: **2 flat faces** and **1 curved surface**, **2 curved edges**, **0 vertices**. A cone: 1 flat face, 1 curved surface, 1 curved edge, 1 vertex (the apex). A sphere: 1 curved surface, no edges, no vertices. Some books count only flat faces; some count the curved surface as a face. There is no single agreed answer, which is one reason the neat counting rules of polyhedra do not apply to curved solids.",{"id":720,"type":53,"title":721,"eyebrow":722,"navLabel":723},"ch08","Nets: solids laid flat","Chapter 08","8 Nets",{"id":725,"type":43,"markdown":726},"nets-precise","A **net** is a flat, connected arrangement of polygons (or other shapes) that folds, along its edges, into the surface of a solid **with no gaps and no overlaps**.\n\nA net must have exactly the right faces: a cube net has 6 equal squares; a square pyramid net has 1 square and 4 identical triangles. But having the right faces is not enough; they must also be joined in the right **arrangement**. Six squares in one long row do **not** fold into a cube, because the ends overlap and two faces are missing on the sides.",{"id":728,"type":62,"caption":729,"columns":730,"rows":733},"table-nets","What the net of each solid contains",[628,731,732],"Pieces in its net","One way to arrange them",[734,737,741,744,747,751,755],[690,735,736],"6 equal squares","A row of 4 with one square above and one below (a cross)",[738,739,740],"Cuboid","6 rectangles in 3 matching pairs","Like the cube cross, with rectangles of the right sizes",[693,742,743],"2 triangles and 3 rectangles","A row of 3 rectangles with a triangle on the top and bottom of the middle one",[647,745,746],"1 square and 4 triangles","A square with a triangle on each side, like a star",[748,749,750],"Tetrahedron","4 equal triangles","A big triangle split into 4, or a row of 4 triangles",[752,753,754],"Cylinder","2 circles and 1 rectangle","The rectangle’s length equals the circles’ circumference",[756,757,758],"Cone","1 circle and 1 sector of a bigger circle","The sector curls round to make the sloping surface",{"id":760,"type":47,"variant":90,"title":761,"markdown":762},"aha-label","Why the cylinder’s rectangle has a fixed length","Peel the label off a tin: it is a rectangle. Its height is the height of the tin, and its length must wrap exactly once round the circular end. So the length of the rectangle is the **circumference** of the circle, about 3.14 × the diameter. A tin 7 cm across needs a label about 22 cm long.",{"id":764,"type":765,"prompt":766,"options":767,"explanation":776},"pred-six-row","prediction","You cut out **6 equal squares in one straight row** and try to fold them into a cube. What happens?",[768,770,772,774],{"id":467,"label":769},"It makes a cube",{"id":470,"label":771},"It makes a cuboid",{"id":473,"label":773},"The ends overlap and two faces are left open",{"id":476,"label":775},"It lies flat and cannot fold","**The ends overlap and two faces are missing.** Folding the row wraps four squares into a tube; squares 5 and 6 land on top of squares 1 and 2. The two open ends of the tube stay uncovered. A cube net needs squares sticking out **sideways** to close the ends. In Investigate you will test many arrangements; exactly 11 of them work.",{"id":778,"type":53,"title":779,"eyebrow":780,"navLabel":781},"ch09","Views from the top, front and side","Chapter 09","9 Views",{"id":783,"type":43,"markdown":784},"views-precise","When architects and engineers draw a building, they use three flat **views**:\n\n- the **top view** or **plan**: looking straight down from above;\n- the **front view** or **front elevation**: looking straight at the front;\n- the **side view** or **side elevation**: looking straight at one side.\n\nEach view flattens the solid into a 2D shape. None of them alone tells you the whole shape, but together they usually do. A **map** is a top view of the land, drawn to scale; that is why roads look like lines and buildings look like rectangles.",{"id":786,"type":106,"title":787,"problem":788,"steps":789},"we-views-stack","Views of a stack of cubes","Four equal cubes are arranged like this: three in a row on the table, and one more on top of the **left** cube. Describe the top, front and side views.",[790,791,792,793],"**Top view:** looking down you see the top of each column. There are 3 columns in a row: **3 squares in a row**.","**Front view:** you see 3 squares along the bottom, with 1 extra square above the left one: an **L shape** of 4 squares.","**Side view (from the right):** the columns hide behind each other, so you see only the tallest: **2 squares stacked**, one on top of the other.","Notice the side view hides 2 cubes completely. That is why builders need **all three** views.",{"id":795,"type":47,"variant":289,"title":796,"markdown":797},"misc-top-count","“The top view shows how many cubes there are”","The top view shows how many **columns** there are, not how many cubes. A single cube and a tower of five cubes have exactly the same top view: one square. To count cubes you must combine the top view (where the columns are) with the front or side view (how tall they are).",{"id":799,"type":53,"title":800,"eyebrow":801,"navLabel":802},"ch10","Line symmetry, precisely","Chapter 10","10 Line symmetry",{"id":804,"type":43,"markdown":805},"sym-precise","A figure has **line symmetry** (or reflection symmetry) if there is a line such that folding along it makes one half land **exactly** on the other. That line is a **line of symmetry** or **axis of symmetry**. Every point on one side has a partner at the same distance on the other side, and the segment joining them crosses the line at a right angle.\n\nSome shapes have one line, some several, some none at all, and a circle has infinitely many.",{"id":807,"type":62,"caption":808,"columns":809,"rows":812},"table-sym-regular","Lines of symmetry of regular polygons",[810,123,313,811],"Regular polygon","Where the lines go",[813,816,818,820,823,826],[814,128,128,815],"Equilateral triangle","Each from a vertex to the midpoint of the opposite side",[316,132,132,817],"2 through opposite vertices, 2 through midpoints of opposite sides",[819,137,137,815],"Regular pentagon",[821,140,140,822],"Regular hexagon","3 through opposite vertices, 3 through midpoints of opposite sides",[824,148,148,825],"Regular octagon","4 through opposite vertices, 4 through opposite midpoints",[76,827,828,829],"(none)","Infinitely many","Every diameter",{"id":831,"type":62,"caption":832,"columns":833,"rows":836},"table-sym-letters","Capital letters and their mirror lines (in a plain font)",[834,835],"Mirror line","Letters",[837,840,843,846],[838,839],"Vertical only","A, M, T, U, V, W, Y",[841,842],"Horizontal only","B, C, D, E, K",[844,845],"Both vertical and horizontal","H, I, O, X",[847,848],"None","F, G, J, L, N, P, Q, R, S, Z",{"id":850,"type":47,"variant":716,"title":851,"markdown":852},"nuance-font","It depends on the font","In a fancy or handwritten font, letters change shape: a curly B or a slanted A may lose its symmetry. The table assumes plain block capitals. Also notice N, S and Z: they have **no** mirror line, but they look the same after a half turn. That is **rotational** symmetry, a different idea you will explore in Extend.",{"id":854,"type":157,"itemId":855,"prompt":856,"check":857,"hints":859,"feedback":861},"pr-oct-sym","shape-and-space.understand-octagon-symmetry","How many lines of symmetry does a regular octagon have?",{"kind":161,"answer":858,"tolerance":163},8,[860],"For a regular polygon, compare the number of lines with the number of sides.",{"correct":862,"incorrect":863},"Yes: a regular polygon with n sides has n lines of symmetry. Octagon: 8.","A regular octagon has 8 lines: 4 through pairs of opposite vertices and 4 through midpoints of opposite sides.",{"id":865,"type":47,"variant":866,"title":867,"markdown":868},"example-taj","example","Symmetry in the Taj Mahal","Stand at the entrance gate of the Taj Mahal and look along the long water channel. Everything on the left, the trees, the fountains, the minarets, the red sandstone buildings, has a matching partner on the right. The whole complex is designed around one great **line of symmetry**. (The only famous break: Shah Jahan's own tomb inside, placed beside Mumtaz Mahal's off the central line.)",{"id":870,"type":871,"title":872,"terms":873},"glossary-understand","glossary","Precise shape vocabulary",[874,877,880,883,886,890,893,897,899,902,905,908,910,912,914,916,918,920,922,924,926,928,931,935,938,941,944,947],{"term":875,"meaning":876},"Line segment","The straight path between two points, including both endpoints.",{"term":878,"meaning":879},"Simple closed curve","A curve that ends where it starts and never crosses itself.",{"term":881,"meaning":882},"Interior \u002F exterior \u002F boundary","The inside, the outside and the curve itself, for a simple closed curve.",{"term":884,"meaning":885},"Adjacent sides","Two sides of a polygon that share a vertex.",{"term":887,"meaning":888,"example":889},"Diagonal","A segment joining two vertices of a polygon that are not next to each other.","A quadrilateral has 2 diagonals.",{"term":891,"meaning":892},"Convex polygon","A polygon with no dents: every diagonal lies inside it.",{"term":894,"meaning":895,"example":896},"Concave polygon","A polygon with at least one interior angle greater than 180°, so it has a dent.","The outline of a star.",{"term":814,"meaning":898},"A triangle with all three sides equal (and so all angles 60°).",{"term":900,"meaning":901},"Isosceles triangle","A triangle with (at least) two equal sides; the angles opposite them are equal.",{"term":903,"meaning":904},"Scalene triangle","A triangle with no two sides equal.",{"term":906,"meaning":907},"Acute \u002F right \u002F obtuse triangle","A triangle whose largest angle is less than, equal to, or greater than 90°.",{"term":335,"meaning":909},"A quadrilateral with a pair of parallel sides.",{"term":330,"meaning":911},"A quadrilateral with both pairs of opposite sides parallel.",{"term":325,"meaning":913},"A quadrilateral with all four sides equal.",{"term":321,"meaning":915},"A quadrilateral with four right angles.",{"term":316,"meaning":917},"A quadrilateral with four equal sides and four right angles.",{"term":341,"meaning":919},"A quadrilateral with two pairs of equal adjacent sides.",{"term":509,"meaning":921},"A segment joining two points on a circle.",{"term":513,"meaning":923},"Part of a circle between two points.",{"term":521,"meaning":925},"The slice of a circle between two radii and an arc.",{"term":517,"meaning":927},"The distance around a circle; about 3.14 times the diameter.",{"term":929,"meaning":930},"Perimeter","The total length of the boundary of a closed figure.",{"term":932,"meaning":933,"example":934},"Polyhedron","A solid whose faces are all flat polygons.","Cube, prism, pyramid.",{"term":936,"meaning":937},"Prism","A polyhedron with two identical parallel ends joined by rectangles.",{"term":939,"meaning":940},"Pyramid","A polyhedron with a polygon base and triangular faces meeting at one apex.",{"term":942,"meaning":943},"Apex","The top point of a pyramid or cone.",{"term":945,"meaning":946},"Plan \u002F elevation","An architect’s top view \u002F front or side view of a building.",{"term":948,"meaning":949},"Axis of symmetry","Another name for a line of symmetry.",{"id":951,"type":952,"title":953,"questions":954},"quiz-understand","quiz","Definitions and properties",[955,968,977,990,999,1008,1021,1034,1044,1057],{"itemId":956,"prompt":957,"options":958,"correct":476,"why":967},"shape-and-space.understand-q-regular","Which of these is a regular polygon?",[959,961,963,965],{"id":467,"label":960},"A rectangle 6 cm by 4 cm",{"id":470,"label":962},"A rhombus with angles 60° and 120°",{"id":473,"label":964},"An isosceles triangle",{"id":476,"label":966},"A square","Regular means equal sides **and** equal angles. The rectangle fails on sides, the rhombus on angles, the isosceles triangle on both. The square passes.",{"itemId":969,"prompt":970,"options":971,"correct":470,"why":976},"shape-and-space.understand-q-diag5","How many diagonals does a pentagon have?",[972,973,974,975],{"id":467,"label":134},{"id":470,"label":137},{"id":473,"label":33},{"id":476,"label":128},"Each vertex has 5 − 3 = 2 diagonals. 5 × 2 = 10 counts each twice, so 10 ÷ 2 = 5.",{"itemId":978,"prompt":979,"options":980,"correct":470,"why":989},"shape-and-space.understand-q-right-eq","Why can no triangle be both equilateral and right-angled?",[981,983,985,987],{"id":467,"label":982},"Equilateral triangles are too small",{"id":470,"label":984},"All angles of an equilateral triangle are 60°",{"id":473,"label":986},"Right angles need two equal sides",{"id":476,"label":988},"It can; it is just rare","Equal sides force equal angles, and three equal angles adding to 180° must be 60° each. None can be 90°.",{"itemId":991,"prompt":992,"options":993,"correct":470,"why":998},"shape-and-space.understand-q-rhombus-diag","The diagonals of a quadrilateral bisect each other at right angles but are not equal. What is its most special name?",[994,995,996,997],{"id":467,"label":316},{"id":470,"label":325},{"id":473,"label":321},{"id":476,"label":335},"Diagonals that bisect each other at 90° mean a rhombus. If they were also equal, it would be a square.",{"itemId":1000,"prompt":1001,"options":1002,"correct":473,"why":1007},"shape-and-space.understand-q-kite","Which quadrilateral has exactly one line of symmetry in general?",[1003,1004,1005,1006],{"id":467,"label":321},{"id":470,"label":330},{"id":473,"label":341},{"id":476,"label":316},"A kite folds along the diagonal joining the vertices between its equal pairs. A rectangle has 2, a general parallelogram has 0, a square has 4.",{"itemId":1009,"prompt":1010,"options":1011,"correct":473,"why":1020},"shape-and-space.understand-q-chord","Which is the longest chord of a circle?",[1012,1014,1016,1018],{"id":467,"label":1013},"Any radius",{"id":470,"label":1015},"An arc",{"id":473,"label":1017},"The diameter",{"id":476,"label":1019},"All chords are equal","A diameter is a chord through the centre, and it is the longest. A radius is not a chord (one end is the centre), and an arc is curved.",{"itemId":1022,"prompt":1023,"options":1024,"correct":470,"why":1033},"shape-and-space.understand-q-perimeter","A rectangular park is 60 m long and 35 m wide. How long is one lap round it?",[1025,1027,1029,1031],{"id":467,"label":1026},"95 m",{"id":470,"label":1028},"190 m",{"id":473,"label":1030},"2,100 m",{"id":476,"label":1032},"130 m","Perimeter = 2 × (60 + 35) = 2 × 95 = 190 m. (2,100 is the area in square metres.)",{"itemId":1035,"prompt":1036,"options":1037,"correct":470,"why":1043},"shape-and-space.understand-q-hexprism","How many edges does a hexagonal prism have?",[1038,1039,1040,1041],{"id":467,"label":637},{"id":470,"label":643},{"id":473,"label":148},{"id":476,"label":1042},"24","6 round each hexagon end (12) plus 6 along the length: 18. In general a prism has 3n edges; 3 × 6 = 18.",{"itemId":1045,"prompt":1046,"options":1047,"correct":476,"why":1056},"shape-and-space.understand-q-net","Which pieces make the net of a cylinder?",[1048,1050,1052,1054],{"id":467,"label":1049},"1 circle and a triangle",{"id":470,"label":1051},"3 circles",{"id":473,"label":1053},"2 rectangles and a circle",{"id":476,"label":1055},"2 circles and a rectangle","The two ends are circles, and the curved surface unrolls into a rectangle whose length is the circumference.",{"itemId":1058,"prompt":1059,"options":1060,"correct":473,"why":1069},"shape-and-space.understand-q-letter","Which letter has both a vertical and a horizontal line of symmetry?",[1061,1063,1065,1067],{"id":467,"label":1062},"A",{"id":470,"label":1064},"B",{"id":473,"label":1066},"H",{"id":476,"label":1068},"S","H is the same on the left and right and the same on top and bottom. A has only a vertical line, B only a horizontal one, and S has none (only rotational symmetry).",{"id":1071,"type":1072,"prompt":1073},"reflect-def","reflection","Write your own definition of a **kite** without looking back. Then test it: does a rhombus pass your test? Does a square? Should they? Compare your definition with the one in this layer and explain any difference.",{"id":1075,"type":1076,"title":1077,"points":1078},"cheat-understand","summary","Cheat sheet",[1079,1080,1081,1082,1083,1084,1085,1086,1087,1088,1089,1090],"A **definition** is the membership test; a **property** is what then follows. Name shapes by definitions.","A **polygon** is a simple closed figure of line segments. **Regular** = all sides equal **and** all angles equal.","A **diagonal** joins non-adjacent vertices. Each vertex has n − 3; total is n(n − 3) ÷ 2. Quadrilateral 2, pentagon 5, hexagon 9, octagon 20.","Triangles by sides: **equilateral, isosceles, scalene**. By largest angle: **acute, right, obtuse**. Angles add to **180°**, so at most one angle is 90° or more.","Family: trapezium ⊃ parallelogram ⊃ rectangle and rhombus ⊃ square. Every square is a rectangle **and** a rhombus. A kite has two pairs of equal adjacent sides.","Diagonals as a test: bisect each other → parallelogram; also equal → rectangle; also perpendicular → rhombus; both → square.","Circle parts: centre, radius, **diameter = 2r** (longest chord), chord, arc, sector, segment, circumference **≈ 3.14 × d**.","**Perimeter** is the distance round. Square 4s, rectangle 2(l + b), regular n-gon n × s. Same perimeter does not mean same area.","Prism with n-sided base: **n + 2 faces, 3n edges, 2n vertices**. Pyramid: **n + 1 faces, 2n edges, n + 1 vertices**.","A **net** needs the right faces **and** the right arrangement. Cylinder net: 2 circles + a rectangle as long as the circumference.","Views: **plan** (top), **front** and **side elevation**. A top view shows columns, not the number of cubes.","A regular n-gon has **n lines of symmetry**. Rectangle 2 (not the diagonals), rhombus 2, kite 1, circle infinitely many.",{"id":1092,"type":294,"conceptId":1093,"relation":296,"explanation":1094},"conn-lines-und","lines","Parallel sides define trapeziums and parallelograms, and perpendicular diagonals identify rhombuses. The Lines topic explains parallel and perpendicular lines.",{"id":1096,"type":294,"conceptId":1097,"relation":1098,"explanation":1099},"conn-hcf-und","hcf-and-lcm","applied_in","The biggest square tile that exactly covers a rectangular floor has a side equal to the HCF of the floor’s length and breadth.",{"id":1101,"type":1102,"sourceIds":1103},"sources-understand","sources",[1104,1105,1106,1107,1108,1109],"shape-and-space-ncert-class6","shape-and-space-ncert-class7","shape-and-space-ncert-class8","shape-and-space-khan-geometry","shape-and-space-mathsisfun-circle","shape-and-space-mathsisfun-quadrilaterals",[1104,1105,1106,1107,1108,1109],"needs_review",{"generatedBy":1113,"notes":1114},"claude-code","Draft generated locally; pending owner review.","dba602c927ed001efa6b348328e1c00f46c4d76c43398b2efb023b41a4f2ed6d",{"logic:practice":1117,"component:sort-game@1":1118,"component:shape-explorer@1":1119,"component:match-pairs@1":1120,"source:shape-and-space-khan-geometry":1121,"source:shape-and-space-mathsisfun-circle":1122,"source:shape-and-space-mathsisfun-quadrilaterals":1123,"source:shape-and-space-ncert-class6":1124,"source:shape-and-space-ncert-class7":1125,"source:shape-and-space-ncert-class8":1126},"3d6b0fe1b15255975a32b0fcd94e8019bc959ad45cbf12e136e86149549c6878","b164f45a2c8ca08f26c450768ff0231e113e9fe45381eddb34dc6d0548596c38","a8965f19a8548e866e5fcd7f4fec4a9adac35ad54d43c9d5c416cdf3348e6198","2a8ee4ac87460b4e1175a4bb13c96b03d577db06dde95670eb7fcfe4ad787899","10f385d6be5a688f5df4a8b9ea08a6c101d7bf683d87c29e7d9d2827a5e11963","ff337c822df6bd0ba5ef54450ce49945d4ef414438058441e656ee73ca685947","292166035f591d3d313675b23dcd587f26add442ae362358b916e3e15caa24b4","e1821bd507663f06793be51247da07646b65ba5a734a31c173a63c03f426f078","20d54bcdc9a5cc5930102f965a7435de619c7bc9731d1de2ff67ae1d0307ca5c","9516472286b1156fa1275e7ea5b6574beff3acc596b8dfe8563f5c446cb7d6b6",{"state":1128,"reviewer":1129,"selfReview":1130,"reviewedAt":1131,"method":1132},"approved","The library owner",true,"2026-09-20T10:18:37.581Z","owner_bulk","preview-7e1cbbcc4f",1789899597021]